Rotation of a spheroid in a Couette flow at moderate Reynolds numbers

Rotation of a spheroid in a Couette flow at moderate Reynolds numbers Zhaosheng Yu Department of Mechanics, Zhejiang University, Hangzhou 310027, China Nhan Phan-Thien Division of Bioengineering, National University of Singapore, Singapore 117576, Singapore Roger I. Tanner School of Aerospace, Mechanical and Mechatronic Engineering, University of Sydney, Sydney NSW 2006, Australia Received 28 March 2007; published 21 August 2007 The rotation of a single spheroid in a planar Couette flow as a model for simple shear flow is numerically simulated with the distributed Lagrangian multiplier based fictitious domain method. The study is focused on the effects of inertia on the orbital behavior of prolate and oblate spheroids. The numerical orbits are found to be well described by a simple empirical model, which states that the rate of the spheroid rotation about the vorticity axis is a sinusoidal function of the corresponding projection angle in the flow-gradient plane, and that the exponential growth rate of the orbit function is a constant. The following transitions in the steady state with increasing Reynolds number are identified: Jeffery orbit, tumbling, quasi-jeffery orbit, log rolling, and inclined rolling for a prolate spheroid; and Jeffery orbit, log rolling, inclined rolling, and motionless state for an oblate spheroid. In addition, it is shown that the orbit behavior is sensitive to the initial orientation in the case of strong inertia and there exist different steady states for certain shear Reynolds number regimes. DOI: 10.1103/PhysRevE.76.026310 PACS number s : 47.15. x I. INTRODUCTION The suspension of nonspherical particles is commonly encountered in engineering applications and the particle orientations are important to the macroscopic behavior of the suspensions. Therefore, the particle orientation has been a subject of extensive investigations. Jeffery 1 analyzed the motion of an ellipsoid theoretically in a simple shear flow at zero Reynolds number, and found that the ellipsoid rotated about the vorticity axis perpendicular to the flow-gradient plane along a closed orbit, dependent solely on the initial condition. Bretherton 2 extended Jeffery s solution to encompass all axisymmetric particles by simply replacing the aspect ratio of particles with an effective aspect ratio. Using asymptotic methods, Harper and Chang 3 showed that the trajectories of a three-dimensional 3D spheroid will be a periodic orbit which corresponds to maximum dissipation. When the effect of fluid elasticity is present, theories 4,5, experiments 6, and numerical simulations 7 revealed that the axis of a fiber or a prolate spheroid spiraled towards the vorticity axis in a weakly elastic fluid. However, in a highly elastic fluid, the final fixed orientation was along the flow direction 8. When the effect of weak fluid inertia is present, both the theory 9 and experiments 10 revealed that the axis of a prolate spheroid gradually turned into the flow-gradient plane, whereas an oblate spheroid eventually aligned its axis of symmetry with the vorticity axis. Feng and Joseph 11 numerically investigated the rotation of an ellipsoid in a simple shear flow and discussed the deviation of the particle motion from the Jeffery orbit due to particle and fluid inertia. Broday et al. 12 examined the motion of spheroid particles in a vertical shear flow, with emphasis on the effects of fluid inertia on the particle migration across streamlines. In these two numerical works, the Reynolds numbers are relatively small. Ding and Aidun 13 investigated numerically the effect of strong fluid inertia on the dynamics of an elliptical cylinder and an ellipsoid suspended in shear flow with the lattice Boltzmann method. The rotational motion was confined to the flow-gradient plane and they found that the angular velocity of the particle decreased with increasing Reynolds number and the rotation stopped when the Reynolds number exceeded a critical value. They found that the rotation period scaled as Re c Re 1/2 near the transitional point; here Re c and Re are the critical Reynolds number and the Reynolds number. Zettner and Yoda 14 studied experimentally the effects of fluids inertia, geometry, and flow confinement upon the dynamics of neutrally buoyant elliptical cylinders over a wide range of aspect ratios in simple shear flow at moderate shear Reynolds numbers, and the results confirmed that an elliptical cylinder of moderate aspect ratio ceases to rotate, resting at a nearly horizontal equilibrium orientation above a critical Reynolds number. The experiment also supported the scaling of Re c Re 1/2. Qi and Luo 15,16 reported some numerical results about the three-dimensional rotations of a spheroid in Couette flow at moderate Reynolds numbers. Their investigations were focused on the transitions in the steady state with increasing Re. They identified three Re regimes for a prolate spheroid: at low Re, the particle tumbled in the flow-gradient plane, as observed previously by Karnis et al. 10 ; at intermediate Re, the axis of the spheroid deviated from the flow gradient, i.e., 0 /2, and at higher Re, the spheroid rolled with its axis parallel to the vorticity axis, i.e., =0. Here, denotes the angle between the orientation of the prolate spheroid and the vorticity axis see Fig. 1. For an oblate spheroid, rolling about the vorticity axis at relatively low Re and inclined rolling at higher Re were identified. The results of Qi and Luo 15,16 enhance greatly the understanding of the 1539-3755/2007/76 2 /026310 11 026310-1 2007 The American Physical Society

YU, PHAN-THIEN, AND TANNER Define as the angle between p and the vorticity Z axis, and as the angle between the projection of p on the X,Y plane and the Y axis see Fig. 1, so that p x = sin sin, p y = sin cos, p z = cos. 3 The solution to the Jeffery equation is tan = a r tan t + k, tan = Ca r ar 2 cos 2 + sin 2, 4 FIG. 1. Schematic diagram of a spheroid rotating in Couette flow. The computational domain is a box with the size of L 1 L 2 L 3. Periodic boundary conditions are introduced in the streamwise X and spanwise Z directions. Note that the definition of in this paper may differ from the definition in some of the literature. rotational and orientational behavior of a spheroid particle in linear shear flow at moderate Reynolds numbers. However, in their simulations, the rotation period of a spheroid did not increase significantly with increasing Reynolds number and an oblate spheroid never ceased to rotate, which are inconsistent with the observations of Ding and Aidun 13, and Zettner and Yoda 14. In the present study, we reexamine the rotation behavior of both prolate and oblate spheroids in Couette flow at moderate Reynolds numbers using the distributed Lagrangian multiplier based fictitious domain DLM/FD method 17,18. Our results are consistent with the observations of Ding and Aidun 13, and Zettner and Yoda 14. The reason for the discrepancies between our results and those of Qi and Luo apart from the methodologies is not known. In addition, we propose a simple empirical model for the prediction of the rotation of a spheroid in simple shear flow, which is described in the next section. In Sec. III, the numerical method is presented. We then report and discuss the results for the prolate and oblate cases, respectively. The concluding remarks will be given in the final section. II. PRELIMINARIES A. Jeffery orbit In a simple shear flow, a prolate spheroid rotates in a Jeffery orbit, described by ṗ = I pp L p, where p is a unit vector along the symmetry axis of the spheroid, L is the effective velocity gradient tensor L = L D, in which L= u T is the velocity gradient tensor, u is the velocity vector, D= L+L T /2 is the strain rate tensor, and =2/ a r 2 +1, where a r is the aspect ratio major diameter/ minor diameter of the spheroid. 1 2 where =2 /T, T=2 a r +a r 1 / is the orbit period, here is the shear rate of the flow. The constants k and C are determined from the initial orientation 0 and 0. B. An empirical model for a general orbit To account for the effects of inertia, we suppose that the motion of the spheroid is still approximated by Jeffery s equation 1, but with the effective velocity gradient tensor L = 0 a 0 b 0 0 0 0 c, 5 where a, b, and c are some constants to be identified using numerical data. The solution to 1 is p = Q/Q, Q x = a /b B0 sin t + k, Q y = B 0 cos t + k, Q z = Q z0 e c t, 6 where = a b, and B0 and k are two constants related to the initial orientation of the spheroid, B 0 = Q 2 2 y0 + b /a Q x0 and tan k = b /a Qx0 /Q y0 = b /a tan 0. To characterize the general orbit, we define an orbit function C t by C t = a 1 r tan t 2 ar cos 2 t + sin 2 t, 7 so that C t is a constant for the Jeffery orbit. From 7, itis clear that tan =C at =m m being an integer and tan =a r C at =m + /2. In light of 3, Eq. 7 can be rewritten as a 2 r p 2 2 y + p x C t =. 8 a r p z To extract the exponential growth rate in C t, we define another orbit function G t =lnc t. 9 With the effective velocity gradient tensor, it is possible for one to establish the constitutive equations of fiber suspensions that account for the effects of fluid inertia or elasticity 19,20. 1. Behavior of the orbit predicted by the model about From Eq. 6 and tan =Q x /Q y, one obtains 026310-2

ROTATION OF A SPHEROID IN A COUETTE FLOW AT tan = a /b tan t + k. Taking the time derivative of 10 yields 10 a = a cos 2 + b sin 2 + b a b = + cos 2, 2 2 11 which means that is a cosine function of with period, average value a +b 2, and amplitude a b 2. For a prolate oblate spheroid, reaches its maximum minimum a at =m and its minimum maximum b at =m + /2. 2. Behavior of the orbit predicted by the model: about G From Eq. 6, we have Q x t + T 4 = a b Q y t, Q z t + T 4 = Q z t e c T/4. Q y t + T 4 = b a Q x t, 12 Then, recalling the definition of C given in 8, one obtains C t + 4 T a 2 r b /a Q 2 x t + a /b Q 2 y t = e C t c T/4. 2 ar Q 2 y t + Q 2 x t 13 For t=t 1 when =m m being an integer, i.e., Q x =0, C t 1 + T 4 C t 1 = 1 a r a b ec T/4, G t 1 + 4 T = G t 1 + c T 4 +ln a b lna r. For t=t 2 when =m + /2, i.e., Q y =0, C t 2 + T 4 C t 2 = a r b a ec T/4, 14 15 16 G t 2 + 4 T = G t 2 + c T 4 ln a b lna r. 17 Equations 15 and 17 imply G = m+1 G =m = G =m + /2 G =m /2 = c T 2. Define A =ln a b lna r. 18 A denotes the average amplitude of G t. G t can be regarded as a combination of a periodic function with period T/2, amplitude A and a linear function of slope c. Given A and T, a and b can be solved, a = 2 a r T ea, b = 2 e A. 19 T a r The Jeffery orbit is recovered at a =a 2 r / 1+a 2 r, b =1/ 1 +a 2 r, and c =0. C. Numerical method The DLM/FD method developed by Glowinski et al. 17 is employed here to numerically simulate the motion of the spheroid. Consider a particle with density d moving in a Newtonian fluid with density l and solvent viscosity. By introducing the scales L c for length, U c for velocity, L c /U c for time, l U 2 c for pressure, l U 2 c /L c for the distributed Lagrange multiplier, d L 3 c for mass, and d L 5 c for moment of inertia, the dimensionless weak formulation of the governing equations in the case of Dirichlet or periodic boundary condition is stated as follows. For t 0, find u W u, p L 2 0, t, U R 3, and R 3 satisfying the following: 1 combined momentum equation, u u t + u vdx p vdx + 1 u T : vdx + r 1 M du Re dt g Frg V d J + =,v V + r P t dt for all v W 0 t, V R 3, R 3 ; 2 incompressibility constraint, 20 q udx = 0 for all q L 2 ; 21 3 rigid-body motion constraint inside (and on) the particle boundary,,u U + r P t = 0 for all t, 22 4 and kinematic equations of the particle, 2 q4 q 1 q 2 1 q 3 q 4 = dx dt = U, q3 q2 q1 q 3 q 4 q 1 q 2 y q 2 q 1 q 4 q 3 z q 1 q 2 q 3 q 4 x 0. 23 24 Here, u, p, U,, and are the fluid velocity, pressure, particle translational and angular velocities, and the distributed Lagrange multiplier, respectively, and v, q, V,, and are their corresponding variations. M and J are the mass and the moment of inertia of the particle, respectively. r is the position vector with respect to the center of the particle X, and g is the acceleration of gravity. In the above equations, 026310-3

YU, PHAN-THIEN, AND TANNER the following dimensionless parameters are introduced: Physical parameters, density ratio r = d / l. Dynamic parameters, 25 Reynolds number Re = lu c L c, 26 Froude number Fr = gl c. 27 U c Here the Froude number represents the relative importance of gravity to inertia. The following variation spaces are used: W u = v H 1 3 v = u t on, W 0 = v H 1 3 v = 0 on, q L2 qdx =0, 28 L 0 2 = and t is an appropriate space by which the constraint of rigid-body motion in P t is enforced. The angle brackets denote an inner product defined in t. P t signifies the region inside and including the particle boundary, is the entire computational domain including both the particle s interior and exterior, and is the boundary of. In 24, q 1,q 2,q 3,q 4 are the components of a quaternion, which can be defined in terms of the Euler angles 21, q 1 = sin /2 cos /2, q 2 = sin /2 sin /2, q 3 = cos /2 sin + /2, q 4 = cos /2 cos + /2. 29 The components are normalized, m q 2 m =1, x, y, and z in 24 represent the components of the angular velocity measured in a body-fixed frame. Both the quaternion and the Euler angles can describe the relationship between the bodyfixed frame and the space-fixed frame. However, the Euler angles themselves cannot be determined unambiguously at sin =0, and consequently are not employed here. With the quaternion, the coordinate transformation matrix from the space-fixed frame to the body-fixed frame is 1 A =2 q 2 + q 2 4 1 2 q 1 q 2 + q 3 q 4 q 1 q 3 q 2 q 4 q 1 q 2 q 3 q 4 q 2 2 + q 2 4 1 2 q 2 q 3 + q 1 q 4. 30 q 1 q 3 + q 2 q 4 q 2 q 3 q 1 q 4 q 2 3 + q 2 4 1 2 Note that A is an orthogonal matrix. For an axisymmetric body such as a spheroid, z axis of a body-fixed frame alone is able to determine the orientation of the body, and that is sin sin, cos sin,cos, or 2 q1 q 3 +q 2 q 4,q 2 q 3 q 1 q 4,q 3 2 +q 4 2 1 2 in the space-fixed frame. D. Computational scheme The operator-splitting technique proposed by Glowinski et al. 17 is used to decompose the system 20 22 into some subsystems. The essential step is to divide the original fluid-solid system into a fluid-flow part and a particle-motion part in the following forms: 1 Fluid-flow part, find u # W u, p L 0 2, satisfying n u# u n + u n u vdx p vdx t + 1 u n+1 T : v n+1 dx = 0 Re for all v W 0 t, 31 q u # dx = 0 for all q L 2. 32 2 Particle-motion part, find u n+1 W u, t, U n+1 R 3, and n+1 R 3 satisfying vdx + r 1 M Un+1 U n Fr g g V un+1 u # t + r 1 J n+1 n t =,v V + r P n, for all v W 0, V R 3, R 3,,u n+1 U n+1 + n+1 r P n = 0 t + n J n 33 for all t, 34 and then determine X n+1 and q i n+1 by integrating 23 and 24. It was found that the finite difference method can be used to solve the above subproblems as a substitute of the finite element method 18. Here, the projection method and the alternating direction implicit ADI technique 18 are used to solve the fluid-flow subproblem. The second part above 33 and 34 is a saddle-point problem and can be solved with the conjugate gradient method 17. The collocation point method is employed to discretize the distributed Lagrange multiplier for a spheroid. The control points are located in a sequence of evenly distributed parallel planes that are perpendicular to the axis of symmetry of the spheroid and their distribution in each plane has the same pattern as devised by Yu et al. 22. The distance between neighboring control points are slightly coarser than the velocity mesh size. In our implementation, the fluid velocity u, the particle translational velocity U, and the Lagrange multiplier are measured in the space-fixed frame, while the particle angular velocity and the position vector r are measured in the body-fixed frame. Therefore, three coordinate transformations are required for solving 33 and 34 : one is for determining P:x=X+A 1 r; one is to transform from the spacefixed frame to the body-fixed frame when dealing with the angular velocity component equation in 33 ; and the last one is to transform r from the body-fixed frame to the spacefixed frame in 34. The reader is referred to 23 for a more detailed description of the algorithm. 026310-4

ROTATION OF A SPHEROID IN A COUETTE FLOW AT The schematic diagram of the flow is given in Fig. 1. The computational domain is a box of sizes L 1 L 2 L 3. Periodic boundary conditions are introduced in the streamwise X and spanwise Z directions. The top and bottom plates move with velocity U and U, respectively. We consider a spheroid of the symmetry axis length 2a and the equatorial diameter 2b. U and L 2 /2 are taken as the characteristic velocity and the characteristic length, respectively. Thus, a time scale is 1/ and the Reynolds number is defined by Re = l UL 2 / 2. We set r =1.001 and Fr=0, as an approximation to the neutrally buoyant case. Such an approximation has been demonstrated to be excellent 18. Actually, it is impossible to encounter a perfectly neutrally buoyant case in practice. The dimensionless control parameters are as follows: confinement ratio, spheroid aspect ratio, a r = a b ; 35 a = 2a L 2 b = 2b L 2 for a prolate spheroid, for an oblate spheroid; shear Reynolds number, Re s = l4a 2 =4 2 a Re for a prolate spheroid, Re s = l4b 2 =4 b 2 Re for an oblate spheroid. 36 37 In addition, because we adopt periodic boundary conditions, the flow fields are also affected by L 1 and L 3. For convenience, we will use L 1 and L 3 to represent the corresponding dimensionless quantities. Throughout this study, we set L 1 =L 2 =2 i.e., a cubic domain, a r =2 and a =0.4 for a prolate spheroid, and a r =0.5 and b =0.4 for an oblate spheroid. We note that the results are not sensitive to the increase in the size of box at a relatively low Reynolds number. We focus on the effects of inertia in the present study, and postpone the effects of other factors such as the interactions between neighboring spheroids and the fluid viscoelasticity to future works. The spheroid is released at the center of the computation domain with zero velocities. Although we did not impose any constraints for the translational motion of the spheroid, the departure of the spheroid center away from the initial position was found negligible for all cases studied. The spatial resolution is 64 64 64, and the time step is 0.005 or 0.01. FIG. 2. Comparison between the calculated orientational orbit and the analytical Jeffery orbit for a prolate spheroid in simple shear flow at a low Reynolds number. Solid lines, our results; dotted lines, analytical results, the two being in excellent agreement. =b/6.4, t=0.001, 0 = /2, and 0 = /4. The two results are in remarkably good agreement with each other. There are no data available for the comparison in the case of 3D and high Reynolds numbers. Therefore, we perform a convergence test for the case of Re s =128, 0 = 6, and 0 = 2, and the results are presented in Fig. 3. The computations at high Reynolds numbers are more difficult than at low Reynolds numbers, and therefore the accuracy of the computational results at high Re appears less accurate compared to the case of low Re. Nevertheless, the results obtained with the mesh of 64 64 64 and the time step of 0.01 converge reasonably well. We will analyze the orbit behavior of a spheroid through the projected rotation angle in the flow-gradient plane t and the orbit function G t. t reflects the behavior of p x and p y, and G t contains the information about p z. Given and G, p x, p y, and p z are uniquely determined. III. RESULTS AND DISCUSSION A. Validation of the numerical method To validate our numerical method, we compare the calculated orientational orbits for a prolate spheroid and the analytical Jeffery orbits in Fig. 2 at a low Reynolds number for a r =2 and 3, respectively. For our computations, Re s =0.5, h FIG. 3. Convergence test at Re s =128, 0 = 6, and 0= 2. 026310-5

YU, PHAN-THIEN, AND TANNER FIG. 4. The growth rate c vs Re s for a prolate spheroid calculated from the data for = 3 2 5 2. 0= 4, 0= 2. FIG. 6. d /dt vs / at different Re s for a prolate spheroid. 0 = 4, 0= 2. B. Prolate case 1. Orbit behavior at different Re We examine the orbit behavior at a fixed initial orientation of 0 = 4 and 0= 2. For Re s=0 256, our numerical results indicate that there exist two critical shear Reynolds numbers and thereby three Reynolds regimes: i Re s =0. The spheroid rotates according to the Jeffery orbit 4. G t is a constant and is a cosine function of with the maximum of 0.8 and the minimum of 0.2 for a r =2. ii Regime one 0 Re s Re sc1 160. Figure 4 reveals that c, the exponential growth rate of the orbit constant C t, is positive, i.e., the prolate spheroid will eventually turn its axis of symmetry into the flow-gradient plane. More precisely, our results show that c is an increasing function of time or G t, and this is more obvious at higher Re s Fig. 5. Thus, we compute c only from G t at = 3 2 5 2 for an overview of the relationship between c and Re s Fig. 4, from which we can see that c reaches its maximum at Re s 40. These results are in qualitative agreement with the experiments of Karnis et al. 10 in which they investigated the behavior of G t at low Re s, and observed that c was approximately a constant with time; its slope with respect to Re s was a decreasing function of Re s. Figure 6 shows that is a cosinelike function of. A further analysis of the behavior of t will be presented later. iii Re s =Re sc1 160. G t is a periodic function with zero growth rate, i.e., c =0. iv Regime two Re sc1 Re s Re sc2. In this regime, the spheroid will turn its axis parallel to the vorticity axis, i.e., c 0. From Fig. 5, the second critical shear Reynolds number Re sc2 is between 224 and 256. At the end of the simulation, the spheroid at Re s =192 is still rotating periodically about the vorticity axis Figs. 5 and 6, whereas, the spheroid at Re s =224 almost ceases the rotation, staying in the vicinity of the flow-vorticity plane Figs. 6 and 7, but continues its approach to the vorticity axis. In this case, the calculations of and G cannot be quantitatively accurate due to two diffi- FIG. 5. The evolutions of the orbit function G at different Re s. 0 = 4, 0= 2. FIG. 7. The evolutions of in cases where the rotation about the vorticity axis of the spheroid stops. 0 = 4, 0= 2. 026310-6

ROTATION OF A SPHEROID IN A COUETTE FLOW AT FIG. 8. tan vs tan t+k at Re s =12.8, as compared to the model prediction. For the simulation result, T=16.6, and for the model prediction, a =0.762, b =0.182. culties: one is that the minimum of is small and the other is that the axis of the spheroid approaches very close to the vorticity axis. There is no meaningful definition for when the two axes coincide exactly. v Regime three Re s Re sc2.asre s increases above Re sc2, the vorticity axis is no longer the stable attractor, and the spheroid stays in almost the flow-vorticity plane with between 0 and 2 Figs. 5 7. Since the spheroid never stops its spin about its axis unless it turns into the flow-gradient plane, we call the motion inclined rolling, the term used by Qi and Luo 15,16. With the shear Reynolds number increasing from zero to 256, the particle s rotation undergoes the following transitions: Jeffery orbit, tumbling, quasi-jeffery orbit, log rolling, and inclined rolling. 2. Assessment of the model We observed that both and G in the simulations exhibit the same characteristics as the model 6. In order to examine to what extent the model can quantitatively predict the orbit, we need to determine the values of a, b, and c from the simulation results and then make a comparison between the two predictions. There are two methods for determining a and b : one is to take a as max and b as min, the other is to calculate them from G with 19. The two methods give the same results if the simulated orbit behavior are in perfect agreement with the model prediction 10 or 11. Inthe following, we only use the former to fit the model to the simulation results, just in order to assess the model. A close scrutiny into Fig. 6 and the corresponding data show that the behavior of is not in perfect agreement with the model. One difference is that the inertial effect results in min occurring at an angle slightly less than m + 2, not precisely at m + 2. This behavior was also observed by Ding and Aidun 13 in their simulations of the rotation of an ellipse in simple shear flow. Figure 6 shows at Re s =12.8, both min and max in all periods are almost the same, being 0.762 and 0.182, respectively. We plot tan as a function of FIG. 9. d /dt min vs Re s for 0 = 6 and 0= 4. The linear fitting function is d /dt min =4.424 10 4 289 Re s. 0 = 2. tan t+k at Re s =12.8 in Fig. 8. It is clear that the result departs slightly away from the model prediction. As a result, it is not surprising that there are discrepancies between the periods obtained with the simulations and the model. The model gives the period of 2 / a b 16.9 using a =0.762 and b =0.182, which is slightly larger than the simulation result of T=16.6. Ding and Aidun 13 found that the rotation period of an elliptical cylinder in simple shear flow was proportional to Re sc Re s 1/2 at small Re sc Re s and they explained the result by two observations: one was that in the vicinity of * where reached its minimum min, min +B * 2, B being a constant; the other was that min Re sc Re s at small Re sc Re s. This result can be easily derived from our model by assuming max is independent of Re s and min is proportional to Re sc Re s, since the model gives T =2 max min 1/2. Figure 6 shows that max is not sensitive to Re s, although it changes with time at relatively high Re s. On the other hand, min approaches zero as Re s increases. The relationship between min and Re s is plotted in Fig. 9. For convenience, min is computed near = 3 2. From Fig. 6, min for different periods does not differ significantly at Re s 192. Accordingly, we calculate the periods using the data at 2, and the result is shown in Fig. 10. Figure 9 illustrates that at Re s from 96 to 256, min decreases approximately linearly with increasing Re s and can be fitted as min =4.424 10 4 289 Re. Simply setting a =0.7 see Fig. 6, we obtain T=357 289 Re s 1/2, which is found systematically a little larger than the periods obtained with the simulations, as demonstrated in Fig. 10. Substituting a =0.7 and b =4.424 10 4 289 Re s into 18 yields the average amplitude A=3 ln 289 Re s /2. This result is compared to the simulation result in Fig. 11 and the model prediction is found systematically smaller than the simulation data. The comparison of the evolutions of G t between the simulation at Re s =38.4 and the model prediction is given in Fig. 12. In the model, a =0.73, b =0.138, and c =0.0267. We can see that apart from small differences in T and A, there is also a phase difference between the two 026310-7

YU, PHAN-THIEN, AND TANNER FIG. 10. The period T vs Re s. For the simulation result, T is calculated from the data for = 2. For the model prediction, a =0.7, b =4.424 10 4 289 Re s. 0 = 4, 0= 2. results due to the fact that the spheroid does not start to rotate from a well-developed orbit, but from a zero angular velocity see Fig. 6, and consequently there is a pronounced transient effect on the rotation of the spheroid at early times that was not captured adequately by the model. The above method for determining a and b matches min and max between the simulation and the model. The other method 19 matches T and A. Due to the slight departure of the orbit characteristics of the spheroid in the simulations away from the model prediction, it is not possible to match all quantities. Since T and A are more important than min and max for the description of the rotation of the spheroid, it may be better to use the second method to determine a and b in the model if one wants to use the model to analyze the rotation of spheroids. FIG. 12. Comparison of the evolutions of G t between the simulation results at Re s =38.4 and the model. For the model, a =0.73, b =0.138, and c =0.0267. 3. Effects of initial orientation Comparison of Fig. 13 and Fig. 5 reveals that the growth rate c is smaller at 0 = 6 than at 0= 4 at the same Re s. Figure 9 also shows that min is typically smaller at 0 = 6 than at 0 = 4 for the same Re s.re sc1 is about 160 for 0 = 4, whereas it is less than 128 for 0= 6. As a result, there exist different steady states at a fixed Reynolds number. For example, at Re s =128, the steady state is log rolling for 0 = 6, but tumbling for 0= 4. The effects of other initial orientations at Re s =128 are plotted in Fig. 14. It seems that the growth rate is lower at a smaller G 0, i.e., smaller 0 and 0 closer to m + 2. We attribute the above bifurcation phenomenon i.e., different steady states at a fixed Reynolds number depending on the initial condition to the particle and fluid inertial effects. At a low Reynolds number, there is no bifurcation since the inertial effect is absent or very small and the rotation of the FIG. 11. The average amplitude A vs Re s for a prolate spheroid, as compared to the model prediction. For the simulation result, A is calculated from G t for = 3 2 5 2. For the model prediction, a =0.7, b =4.424 10 4 289 Re s. 0 = 4, 0= 2. FIG. 13. The evolutions of the orbit function G at different Re s. 0 = 6, 0= 2. 026310-8

ROTATION OF A SPHEROID IN A COUETTE FLOW AT FIG. 14. The evolutions of the orbit function G at different initial orientations. Re s =128. spheroid solely depends on the hydrodynamic force. By contrast, the particle inertia plays an important role in the motion of the spheroid at a high Reynolds number, and therefore the spheroid released from different orientations at initial time can follow different orientational orbits and reach different steady states. C. Oblate case We first examine the orbit behavior at a fixed initial orientation of 0 = 4 and 0=0. For Re s =0 192, the numerical results indicate that there exist three Reynolds regimes for different steady states: i Regime one 0 Re s Re sc1. The oblate spheroid will eventually turn its axis of symmetry parallel to the vorticity axis. As shown in Fig. 15, the turning velocity for the oblate case is much higher than that for the prolate case at the comparable Re s. Persistent rotation of the spheroid about the vorticity axis is observed at Re s =12.8 and Re s =38.4, and FIG. 16. d /dt vs / at Re s =12.8 and 38.4 for an oblate spheroid. 0 = 4, 0=0. Fig. 16 reveals that is a sinelike function of, still approximately satisfying the model prediction. min occurs at angles slightly larger than m, and this is an anomalous behavior, because we have seen that the orientation of the broadside of the prolate spheroid is slightly less than m + 2, when reaches its minimum, and we will see that the fixed orientation of the broadside of the oblate spheroid is also slightly less than m + 2 when the oblate spheroid completely ceases its rotation about the vorticity axis at high Re s Fig. 18. Figure 17 shows that the regularly periodic rotations will stop at Re s =64 and Re s =89.6. For Re s =64, the rotation can pass through = and our data show min occurs at an angle larger than. For Re s =89.6, the spheroid rotates back soon after its release and stays at the orientation of slightly smaller than zero for a long time. The orientation data of the spheroids at t 60 for both Re s =64 and Re s =89.6 shown in Fig. 17 may not be reliable because at that time p x and p y are too small see Fig. 15. ii Regime two Re sc1 Re s Re sc2. When Re s increases up to 128, the spheroid will not align its axis with the vor- FIG. 15. The evolutions of the orbit function G at different Re s for an oblate spheroid. 0 = 4, 0=0. FIG. 17. The evolutions of / at Re s =64 and 89.6 for an oblate spheroid. 0 = 4, 0=0. 026310-9

YU, PHAN-THIEN, AND TANNER FIG. 18. The evolutions of / at Re s =128, 160, and 192 for an oblate spheroid. 0 = 4, 0=0. FIG. 20. The evolutions of the orbit function G at different Re s for an oblate spheroid. 0 = 3, 0=0. ticity axis, but stays almost in the gradient-vorticity plane is about 0.06, from Fig. 18, with between 0 and 2. This is the state of inclined rolling. It is difficult to determine the subcritical and supercritical Reynolds numbers Re sc1 and Re sc2, since G or cannot be calculated accurately when the spheroid axis is very close to the vorticity axis or the flowgradient plane. For example, it is difficult to discern if Re s = 89.6 belongs to the log-rolling regime or the inclinedrolling regime, based on the result in Fig. 15 and in Fig. 19. It may be more suitable to define a critical G or threshold for demarcating this regime. Figures 15 and 19 show that the Reynolds number interval for an obvious inclined-rolling state is small, and is expected to be much narrower than 89.6,160. iii Regime three Re s Re sc2. In this regime, the oblate spheroid will turn its axis of symmetry parallel to the gradient axis along approximately the gradient-vorticity plane. Figure 18 shows that the deviation of the orientation from the gradient axis m increases with increasing Re s, consistent with the results of Ding and Aidun 13 for the ellipse case. In addition, Fig. 19 indicates that there is an overshoot for the spheroid approaching the flow-gradient plane, resulting in a striking pulse of G in Fig. 15. Therefore, with the shear Reynolds number increasing from zero to 192, the rotation of an oblate spheroid undergoes the following transitions: Jeffery orbit, log rolling, inclined rolling, and motionless state. The log-rolling and inclined-rolling states have also been identified by Qi and Luo 15,16. The results of G at 0 = 3 and 0=0 are shown in Fig. 20. Like the prolate case, the behavior of G also depends on the initial orientations of the spheroid. The significant effects of the initial orientations occur at Re s being 128 and larger. In this high Re s regime, it appears that the axis of the spheroid for 0 = 3 can approach closer to the vorticity axis and then the turning of the axis of the spheroid away from the vorticity axis is hindered, compared to the case of 0 = 4 Figs. 15 and 20. Again, at Re s =128, there exist two steady states: log rolling for 0 = 3 and inclined rolling for 0= 4. IV. CONCLUSIONS FIG. 19. The evolutions of p z at Re s =128, 160, and 192 for an oblate spheroid. 0 = 4, 0=0. We have presented the DLM/FD algorithm for the simulation of the motion of three-dimensional nonspherical particles in fluids and a simple empirical model for the description of the rotation of a spheroid in simple shear flow and possibly other types of flows. The numerical results on the rotation of both prolate and oblate spheroids in a Couette flow have been reported and compared to the empirical model. The following conclusions can be drawn from the present study: 1 When the effects of inertia are not too strong to cause the stop of the rotation of the spheroid axis about the vorticity axis, is approximately a cosine function of and the 026310-10

ROTATION OF A SPHEROID IN A COUETTE FLOW AT exponential grow rate of the orbit function c is approximately a constant. These two characteristics of the orbit can be well predicted by a simple model. 2 For a fixed initial orientation, with Re s increasing from 0 to 256, the steady states for a prolate spheroid undergo the following transitions: Jeffery orbit, tumbling, quasi-jeffery orbit, log rolling, and inclined rolling. 3 For a fixed initial orientation, with Re s increasing from 0 to 192, the steady states for an oblate spheroid undergo the following transitions: Jeffery orbit, log rolling, inclined rolling, and motionless state. 4 The orbit behavior is significantly affected by the initial orientation. There exist different steady states for certain shear Reynolds number regimes. ACKNOWLEDGMENTS The authors wish to thank Professor Xijun Fan for helpful discussions about the empirical model. The work is mainly based on a part of the thesis work of one of the authors Z.Y. at the University of Sydney. 1 G. B. Jeffery, Proc. R. Soc. London, Ser. A 102, 161 1922. 2 F. P. Bretherton, J. Fluid Mech. 14, 284 1962. 3 E. Y. Harper and I.-D. Chang, J. Fluid Mech. 33, 209 1968. 4 L. G. Leal, J. Fluid Mech. 69, 305 1975. 5 O. G. Harlen and L. L. Koch, J. Fluid Mech. 252, 187 1993. 6 Y. Iso, D. L. Koch, and C. Cohen, J. Non-Newtonian Fluid Mech. 62, 115 1996. 7 N. Phan-Thien and X. J. Fan, J. Non-Newtonian Fluid Mech. 105, 131 2002. 8 Y. Iso, C. Cohen, and D. L. Koch, J. Non-Newtonian Fluid Mech. 62, 135 1996. 9 P. G. Saffman, J. Fluid Mech. 1, 540 1956. 10 A. Karnis, H. L. Goldsmith, and S. G. Mason, Can. J. Chem. Eng. 44, 181 1966. 11 J. Feng and D. D. Joseph, J. Fluid Mech. 303, 83 1995. 12 D. Broday, M. Fichman, M. Shapiro, and C. Gutfinger, Phys. Fluids 10, 86 1998. 13 E.-J. Ding and C. K. Aidun, J. Fluid Mech. 423, 317 2000. 14 C. M. Zettner and M. Yoda, J. Fluid Mech. 442, 241 2001. 15 D. W. Qi and L. S. Luo, Phys. Fluids 14, 4440 2002. 16 D. W. Qi and L. S. Luo, J. Fluid Mech. 477, 201 2003. 17 R. Glowinski, T.-W. Pan, T. I. Hesla, and D. D. Joseph, Int. J. Multiphase Flow 25, 755 1999. 18 Z. Yu, N. Phan-Thien, and R. I. Tanner, J. Fluid Mech. 518, 61 2004. 19 F. P. Folgar and C. L. Tucker, J. Reinf. Plast. Compos. 3, 98 1984. 20 X. J. Fan, Ph.D. dissertation, University of Syndey, 2006. 21 D. C. Rapaport, The Art of Molecular Dynamics Simulation Cambridge University Press, Cambridge, 1995. 22 Z. Yu, N. Phan-Thien, Y. Fan, and R. I. Tanner, J. Non- Newtonian Fluid Mech. 104, 87 2002. 23 Z. Yu, Ph.D. dissertation, University of Sydney, 2003. 026310-11